There are probably more than a dozen papers available online that give proofs of Dirichlet's theorem on primes in arithmetic progression and the derivation of the class number formula. google will help you find one that suits your needs (e.g. people.reed.edu/~jerry/361/lectures/iqclassno.pdf, to give but one source found within 1 sec). Davenport's Multiplicative number theory is a good start if you like books. Fröhlich and Taylor have a book on algebraic number theory dealing with class number formulas.
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Franz LemmermeyerJul 5 '11 at 17:27

2 Answers
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Where you might want to start: The classical approach is based on special functions, and given e.g. here: http://www-math.mit.edu/~kedlaya/Math254B/zetafunction.pdf (I found this directly with google). I think the standard reference for such things is Neukirch "Algebraic Number Theory" and the later chapters on $L$ functions in this text.

A more elegant point of view: Tate's thesis gives the modern picture, but it is not free available, e.g. it is the last chapter in Cassels & Fr\"{o}hlich - Algberaic number theory. It is quiet self contained and very pleasant to read, if you know the basics about the Fourier transform of an locally compact abelian group. To learn the Fourier analysis, I recommend the first chapter of Rudin - Fourier analysis on groups as a start, and to translate every statement to the locally compact group $\mathbb{R}$ to get a good idea, what is going on. I think that Tate's approach is much more enlightening than the classical one, and there are many people which have rewritten parts of his thesis in various lecture notes, which are freely available online (use google). The key point of Tate's interpretation is that the class number formula is interpreted as a certain volume, and all classical functions, which turn up in the classical arguments, arise more naturally.

Tate's thesis is not an introduction to Dirichlet's class number formula, and most certainly not very nice to read for the one who asked the question.
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Franz LemmermeyerJul 5 '11 at 17:23

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But it gives the "right" interpretation as a first instance of a Tamagawa number, which should be what the OP was searching for, if you consider his other questions. I edited the question to put that right.
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Marc PalmJul 5 '11 at 18:53

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I strongly agree with Franz Lemmermeyer. You'll never be happy with interpreting the class number as a volume if you don't cut your teeth on the more classical point of view.
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David HansenJul 5 '11 at 21:05

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Different learners differ on this point, but I find most students do better when they see a concrete presentation first and an elegant one second. Tate's thesis is extremely elegant, but scores low on concreteness. So I would not recommend that as a place to start.
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David SpeyerJul 6 '11 at 20:13

Well, I don't know any reference that examines ''each nook and corner'', but as Kevin says in a comment above, Washington's ''Cyclotomic fields'' is one good place to start (assuming you have enough grounding in algebraic number theory).

In addition, Lang's ''Algebraic number theory'' contains some things on class number formulas I think; also we have his two-volume (now as one-volume at Springer) book(s) on Cyclotomic fields. But beware, I would say that these/this require a firmer background, than Washington's book.

(Without encouraging illegal activities, I'm sure there are some bootleg versions on the net of the above books.)

Otherwise, there are dozens of nice books on the subject. Also, there must be tonnes of free lecture notes out there in cyberspace.